Vertex deletion parameterized by elimination distance and even less.

We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · nO(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · nO(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. The first class of parameterizations is based on the notion of elimination distance of the input graph to the target graph class , which intuitively measures the number of rounds needed to obtain a graph in by removing one vertex from each connected component in each round. The second class of parameterizations consists of a relaxation of the notion of treewidth, allowing arbitrarily large bags that induce subgraphs belonging to the target class of the deletion problem as long as these subgraphs have small neighborhoods. Both kinds of parameterizations have been introduced recently and have already spawned several independent results. Our contribution is twofold. First, we present a framework for computing approximately optimal decompositions related to these graph measures. Namely, if the cost of an optimal decomposition is k, we show how to find a decomposition of cost kO(1) in time f(k) · nO(1). This is applicable to any class for which we can solve the so-called separation problem. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations. For example, we present an algorithm running in time nO(1) + 2kO(1)·(n+m) and polynomial space for Odd cycle transversal parameterized by the elimination distance k to the class of bipartite graphs.